The line segment between two points is the shortest.

The complementary angles of the same angle or equal angle are equal.

The complementary angles of the same angle or the same angle are equal.

One and only one straight line is perpendicular to the known straight line.

Of all the line segments connecting a point outside the straight line with points on the straight line, the vertical line segment is the shortest.

7 The axiom of parallelism passes through a point outside a straight line, and there is only one straight line parallel to this straight line. If two straight lines are parallel to the third straight line, the two straight lines are also parallel to each other.

The same angle is equal and two straight lines are parallel.

The internal dislocation angles of 10 are equal, and the two straight lines are parallel.

1 1 are complementary and two straight lines are parallel.

12 Two straight lines are parallel and have the same angle.

13 two straight lines are parallel, and the internal dislocation angles are equal.

14 Two straight lines are parallel and complementary.

Theorem 15 The sum of two sides of a triangle is greater than the third side.

16 infers that the difference between two sides of a triangle is smaller than the third side.

The sum of the internal angles of 17 triangle is equal to 180.

18 infers that the two acute angles of 1 right triangle are complementary.

19 Inference 2 An outer angle of a triangle is equal to the sum of two non-adjacent inner angles.

Inference 3 The outer angle of a triangle is greater than any inner angle that is not adjacent to it.

2 1 congruent triangles has equal sides and angles.

The axiom of 22 angles (SAS) has two triangles with equal angles. The axiom of 23 angles (ASA) has two angles with equal sides. 24 Inference (AAS) has two angles, and the opposite sides of one angle are equal. The axiom of 25 sides (SSS) has two triangles with equal sides.

Axiom of hypotenuse and right angle (HL) Two right angle triangles with hypotenuse and a right angle are congruent. Theorem 1 The distance between a point on the bisector of an angle and both sides of the angle is equal.

Theorem 2 is a point with equal distance on both sides of an angle. On the bisector of this angle, the bisector of angle 29 is the set of all points with equal distance to both sides of the angle.

Theorem of the nature of isosceles triangle 30 The two base angles of isosceles triangle are equal (that is, equilateral angles) 3 1 inference 1 The bisector of the vertices of isosceles triangle bisects the base and is perpendicular to the base.

The bisector of the top angle, the median line on the bottom edge and the height on the bottom edge of the isosceles triangle coincide with each other.

34 Judgment Theorem of an isosceles triangle If a triangle has two equal angles, then the opposite sides of the two angles are also equal (equal angles and equal sides).

Inference 1 A triangle with three equal angles is an equilateral triangle.

Inference 2 An isosceles triangle with an angle equal to 60 is an equilateral triangle.

In a right triangle, if an acute angle is equal to 30, the right side it faces is equal to half of the hypotenuse.

The center line of the hypotenuse of a right triangle is equal to half of the hypotenuse.

Theorem 39 The distance between the point on the vertical line of a line segment and the two endpoints of the line segment is equal.

40 The inverse theorem is the point where the distance between the two endpoints of a line segment is equal. On the midline of this line segment, the midline of the line segment 4 1 can be regarded as the set of all points with the same distance between the two endpoints of this line segment.

Theorem 2: If two figures are symmetrical about a straight line, then the symmetry axis is the perpendicular to the straight line connecting the corresponding points.

Theorem 3 Two graphs are symmetrical about a straight line. If their corresponding line segments or extension lines intersect, then the intersection point is on the axis of symmetry.

45 Inverse Theorem If the straight line connecting the corresponding points of two graphs is bisected vertically by the same straight line, then the two graphs are symmetrical about this straight line.

46 Pythagorean Theorem The sum of squares of two right angles A and B of a right triangle is equal to the square of the hypotenuse C, that is, A 2+B 2 = C 2.

47 Inverse Theorem of Pythagorean Theorem If the three sides of a triangle A, B and C are related in length A 2+B 2 = C 2, then the triangle is a right triangle.

The sum of the quadrilateral internal angles of Theorem 48 is equal to 360.

The sum of the external angles of the quadrilateral is equal to 360.

Theorem of the sum of internal angles of 50 polygons: Is the sum of internal angles of n polygons equal to (n-2)? 180

5 1 It is inferred that the sum of the external angles of any polygon is equal to 360.

52 parallelogram property theorem 1 parallelogram diagonal equality

53 parallelogram property theorem 2 The opposite sides of parallelogram are equal

It is inferred that the parallel segments sandwiched between two parallel lines are equal.

55 parallelogram property theorem 3 diagonal bisection of parallelogram.

56 parallelogram decision theorem 1 Two groups of parallelograms with equal diagonals are parallelograms 57 parallelogram decision theorem 2 two groups of parallelograms with equal opposite sides are parallelograms 58 parallelogram decision theorem 3 parallelograms bisected diagonally are parallelograms.

59 parallelogram decision theorem 4 A group of parallelograms with equal opposite sides are parallelograms 60 Rectangular property theorem 1 All four corners of a rectangle are right angles.

6 1 rectangle property theorem 2 The diagonals of rectangles are equal

62 Rectangular Decision Theorem 1 A quadrilateral with three right angles is a rectangle.

63 Rectangular Decision Theorem 2 Parallelograms with equal diagonals are rectangles

64 diamond property theorem 1 all four sides of the diamond are equal.

65 Diamond Property Theorem 2 Diagonal lines of diamonds are perpendicular to each other, and each diagonal line bisects a set of diagonal lines. 66 diamond area = half of diagonal product, that is, S=(a? b)2

67 diamond decision theorem 1 A quadrilateral with four equilateral sides is a diamond.

68 Diamond Decision Theorem 2 Parallelograms whose diagonals are perpendicular to each other are diamonds.

69 Theorem of Square Properties 1 All four corners of a square are right angles and all four sides are equal.

Theorem of 70 Square Properties 2 Two diagonal lines of a square are equal and bisected vertically, and each diagonal line bisects a set of diagonal lines.

Theorem 7 1 1 is congruent with respect to two centrosymmetric graphs.

Theorem 2 About two graphs with central symmetry, the connecting lines of symmetric points both pass through the symmetric center and are equally divided by the symmetric center.

Inverse Theorem If a straight line connecting the corresponding points of two graphs passes through a point and is bisected by the point, then the two graphs are symmetrical about the point.

The property theorem of isosceles trapezoid is that two angles of isosceles trapezoid on the same base are equal.

The two diagonals of an isosceles trapezoid are equal.

Decision theorem of isosceles trapezoid 76 A trapezoid with two equal angles on the same base is an isosceles trapezoid 77 A trapezoid with equal diagonal lines is an isosceles trapezoid.

Theorem of parallel lines bisecting line segments If a group of parallel lines have the same line segment on a straight line, so do the line segments on other straight lines.

79 Inference 1 A straight line passing through the midpoint of one waist of a trapezoid and parallel to the bottom will bisect the other waist.

Inference 2 A straight line passing through the midpoint of one side of a triangle and parallel to the other side will bisect the third side.

The median line theorem of 8 1 triangle The median line of a triangle is parallel to the third side and equal to half of it.

The trapezoid midline theorem is parallel to the two bases and equal to half of the sum of the two bases. L=(a+b)÷2 S=L? h

Basic properties of ratio 83 (1) If a:b=c:d, then ad=bc.

If ad=bc, then a: b = c: d.

84 (2) Combinatorial Properties If A/B = C/D, then (A B)/B = (C D)/D.

85 (3) Isometric Property If A/B = C/D =? =m/n(b+d+？ +n 0), then

(a+c+？ +m)/(b+d+？ +n)=a/b

86 parallel lines are divided into segments and the theorem of proportionality. Three parallel lines cut two straight lines, and the corresponding segments are proportional.

It is inferred that the line parallel to one side of the triangle cuts the other two sides (or the extension lines of both sides), and the corresponding line segments are proportional.

Theorem 88 If the corresponding line segments obtained by cutting two sides (or extension lines of two sides) of a triangle are proportional, then this straight line is parallel to the third side of the triangle.

A straight line parallel to one side of a triangle and intersecting with the other two sides, the three sides of the cut triangle are directly proportional to the three sides of the original triangle.

Theorem 90 A straight line parallel to one side of a triangle intersects the other two sides (or extension lines of both sides), and the triangle formed is similar to the original triangle.

9 1 similar triangles's decision theorem 1 Two angles are equal and two triangles are similar (ASA)

The right triangle is divided into two right triangles according to the height on the hypotenuse, which is similar to the original triangle. Decision Theorem 2: The two sides are in direct proportion and the included angle is equal. Decision Theorem 3: Three sides are proportional and two triangles are similar (SSS).

Theorem 95 If the hypotenuse and right-angled side of one right-angled triangle are proportional to the hypotenuse and right-angled side of another right-angled triangle, then the two right-angled triangles are similar. Theorem 96 The ratio of similar triangles to is 1, and the ratio of the corresponding median line to the corresponding bisector is equal to the similarity ratio.

97 Property Theorem 2 The ratio of similar triangles perimeter is equal to similarity ratio.

98 Property Theorem 3 The ratio of similar triangles area is equal to the square of similarity ratio.

The sine of any acute angle is equal to the cosine of the remaining angles, and the cosine of any acute angle is equal to the sine of the remaining angles.

100 The tangent of any acute angle is equal to the cotangent of other angles, and the cotangent of any acute angle is equal to the tangent of other angles.

10 1 A circle is a set of points whose distance from a fixed point is equal to a fixed length.

102 The interior of a circle can be regarded as a collection of points whose center distance is less than the radius.

The outer circle of 103 circle can be regarded as a collection of points whose center distance is greater than the radius.

104 The radius of the same circle or equal circle is the same.

The distance from 105 to the fixed point is equal to the trajectory of a fixed-length point, which is a circle with the fixed point as the center and the fixed length as the half diameter.

106 and the locus of the point with the same distance between the two endpoints of the known line segment is the middle vertical line of the line segment.

The locus from 107 to a point with equal distance on both sides of a known angle is the bisector of this angle.

The trajectory from 108 to the equidistant point of two parallel lines is a straight line parallel and equidistant to these two parallel lines.

Theorem 109 Three points that are not on the same straight line determine a circle.

1 10 vertical diameter theorem divides the chord perpendicular to its diameter into two parts, and divides the two arcs opposite to the chord into two parts.

1 1 1 inference 1 ① bisect the diameter of the chord (not the diameter) perpendicular to the chord, and bisect the two arcs opposite to the chord through the center of the circle.

③ bisect the diameter of an arc opposite to the chord, bisect the chord vertically, and bisect another arc opposite to the chord 1 12. It is inferred that the arcs sandwiched by two parallel chords of a circle are equal.

1 13 circle is a centrosymmetric figure with the center of the circle as the symmetry center.

Theorem 1 14 In the same circle or in the same circle, the isocentric angle has equal arc, chord and chord center distance.

1 15 It is inferred that in the same circle or equal circle, if one set of quantities in two central angles, two arcs, two chords or the chord-center distance between two chords is equal, the corresponding other set of quantities is also equal.

Theorem 1 16 The angle of an arc is equal to half its central angle.

1 17 Inference 1 The circumferential angles of the same arc or the same arc are equal; In the same circle or in the same circle, the arcs of equal circumferential angles are also equal.

1 18 Inference 2 The circumferential angle (or diameter) of a semicircle is a right angle; 90 degree circle angle

The chord on the right is the diameter.

1 19 Inference 3 If the median line of one side of a triangle is equal to half of this side, then this triangle is a right triangle.

120 Theorem The diagonals of the inscribed quadrilateral of a circle are complementary, and any external angle is equal to its internal angle.

12 1① the intersection of the straight line l and ⊙O is d < r.

(2) the tangent of the straight line l, and ⊙ o d = r.

③ lines l and ⊙O are separated by d > r.

122 A theorem for determining that a straight line passing through the outer end of the radius and perpendicular to this radius is a tangent of a circle. 123 The property theorem of tangent line The tangent line of a circle is perpendicular to the radius passing through the tangent point.

124 Inference 1 A straight line passing through the center of the circle and perpendicular to the tangent must pass through the tangent point.

125 Inference 2 A straight line passing through the tangent and perpendicular to the tangent must pass through the center of the circle.

The tangent length theorem 126 leads to two tangents of a circle from a point outside the circle, and their tangent lengths are equal. The line between the center of the circle and this point bisects the included angle of the two tangents.

127 The sum of two opposite sides of a circle's circumscribed quadrilateral is equal.

128 Chord Angle Theorem The chord angle is equal to the circumferential angle of the arc pair it clamps.

129 it is inferred that if the arcs sandwiched by two chord tangent angles are equal, then the two chord tangent angles are also equal. 130 intersection chord theorem The length of two intersecting chords in a circle divided by the product of the intersection point is equal.

13 1 Inference: If the chord intersects the diameter vertically, then half of the chord is formed by dividing it by the diameter.